Finite Group Databasebeta

← #329467

Subgroups — #329467

Order 512.

690 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic
2#29664C212yesmaximal
3#21300C211yesmaximal, characteristic
4#10617C212yesmaximal, complemented
5#10479C214yesmaximal, complemented
6#10265C212yesmaximal, complemented
7#9137C212yesmaximal
8#8595C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#500C2211yescharacteristic
11C2×#462C2214yes—
12C2×#423C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2857C2212yes—
15#2024—22no—
16#2016—22no—
17#2001—24no—
18#1990C2212yes—
19#1984—21no—
20#1984—21no—
21#1984C2212yes—
22#1984C2212yes—
23#1942—21no—
24#1942—21no—
25#1845C2211yescharacteristic
26#1808—21no—
27#1808—21no—
28#1803C2214yes—
29#1797C2212yes—
30#1729—21no—
31#1729—21no—
32C2×#334C2212yes—
33C2×#332C2214yes—
34#1296C2214yes—
35#1277C2212yes—
36C22×Q16—24no—
37C22×M16—21no—
38C23×C8D411yescharacteristic
39C23×C8D411yescharacteristic
40#533—42no—
41#533—42no—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8C2312yes—
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8D412yes—
49C2×C4×Q8—21no—
50C2×C4×Q8C2312yes—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#500—42no—
54#462—44no—
55#462—44no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107—21no—
60#423—42no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2314yes—
64C2×#103—24no—
65C2×#103—24no—
66C2×C22⋊C8—22no—
67C2×C4×C8—42no—
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#391—22no—
73#389—22no—
74#389—22no—
75#384C2312yes—
76#380—22no—
77#380—22no—
78#340—22no—
79#340—22no—
80#340—24no—
81#336—22no—
82#336—22no—
83#336—22no—
84#336—22no—
85#336—22no—
86#334—42no—
87#332—44no—
88C22×Q8—21no—
89C22×Q8—22no—
90C22×Q8C2×D412yes—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C2×Q16—48no—
95C2×Q16—84no—
96C2×Q16—84no—
97C2×M16—42no—
98C2×M16—42no—
99C2×M16—42no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—24no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8Q1614yescomplemented
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C4⋊Q8—24no—
113C4⋊Q8—24no—
114C4⋊Q8—42no—
115C4⋊Q8—42no—
116#125—44no—
117C22⋊Q8—44no—
118C22⋊Q8—44no—
119C22⋊Q8—84no—
120C4×Q8—24no—
121C4×Q8—24no—
122C4×Q8—42no—
123C4×Q8—42no—
124#117—42no—
125#117—42no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C2×D412yes—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—42no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42C2411yescharacteristic
147#107—42no—
148#107—42no—
149#107—42no—
150#107—42no—
151C4⋊C8—24no—
152C4⋊C8—24no—
153C4⋊C8—24no—
154C4⋊C8—42no—
155#103—48no—
156#103—84no—
157C22⋊C8—82no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—41no—
161C4×C8—41no—
162#95—42no—
163#95—42no—
164C24#12111yescharacteristic
165C2×Q8—24no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—24no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—42no—
175C2×Q8—42no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4C2×Q1612yes—
194C22×C4#13712yes—
195C22×C4C22×D411yescharacteristic
196C22×C4C22×D411yescharacteristic
197C22×C4C2×#4111yescharacteristic
198Q16—816no—
199M16—84no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—44no—
216C2×C8—44no—
217C2×C8C2×Q1612yes—
218C2×C8C2×Q1612yes—
219C2×C8#13712yes—
220C4⋊C4—24no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C22⋊C4—44no—
235C22⋊C4—84no—
236C42—21no—
237C42—22no—
238C42—21no—
239C42—22no—
240C42—42no—
241C42—42no—
242C42H(5,𝔽2)12yes—
243C42H(5,𝔽2)12yes—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#46212yes—
251C23C2×#12111yescharacteristic
252Q8—44no—
253Q8—44no—
254Q8—84no—
255Q8—84no—
256Q8—84no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4#46212yes—
285C2×C4#47312yes—
286C2×C4#53411yescharacteristic
287C2×C4#53412yes—
288C2×C4#53511yescharacteristic
289C2×C4#53812yes—
290C2×C4C22×Q1611yescharacteristic
291C2×C4C2×#13711yescharacteristic
292C2×C4C2×#13711yescharacteristic
293C2×C4#57511yescharacteristic
294C2×C4#57612yes—
295C2×C4#57712yes—
296C8—22no—
297C8—22no—
298C8—22no—
299C8—41no—
300C8—84no—
301C8—82no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#261711yescharacteristic
310C22C2×#46211yescharacteristic
311C22#303711yescharacteristic
312C22#305811yescharacteristic
313C22#306011yescharacteristic
314C22C2×#47311yescharacteristic
315C22#306711yescharacteristic
316C4—22no—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—82no—
321C4—82no—
322C4—162no—
323C4—162no—
324C4#318911yescharacteristic
325C4#319511yescharacteristic
326C4#319814yes—
327C4#320911yescharacteristic
328C4#321511yescharacteristic
329C2—42no—
330C2#1920811yescharacteristic
331C2#1921011yescharacteristic
332C2#1934111yescharacteristic
333C2#1935011yescharacteristic
334C2#2971511yescharacteristic
335C2#2973611yescharacteristic
336C2#2974911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #10617C2semidirect
5. #10479C2semidirect
6. #10265C2semidirect
107. C22×C8Q16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#332092
#296642
#213002
#106172
#104792
#102652
#91372
#85952